Coherence of a Driven Electron Spin Qubit Actively Decoupled from Quasistatic Noise

The coherence of electron spin qubits in semiconductor quantum dots suffers mostly from low-frequency noise. During the past decade, efforts have been devoted to mitigate such noise by material engineering, leading to substantial enhancement of the spin dephasing time for an idling qubit. However, the role of the environmental noise during spin manipulation, which determines the control fidelity, is less understood. We demonstrate an electron spin qubit whose coherence in the driven evolution is limited by high-frequency charge noise rather than the quasistatic noise inherent to any semiconductor device. We employ a feedback-control technique to actively suppress the latter, demonstrating a $\pi$-flip gate fidelity as high as $99.04\pm 0.23\%$ in a gallium arsenide quantum dot. We show that the driven-evolution coherence is limited by the longitudinal noise at the Rabi frequency, whose spectrum resembles the $1/f$ noise observed in isotopically purified silicon qubits.

Popular Summary

Future quantum devices will require precise control of their building blocks: quantum bits (or qubits). For qubits that encode information in electron spins (spin qubits), control precision is getting better thanks to steady improvements in maintaining the coherence of the quantum state. Researchers have explored extending coherence by dynamically flipping the qubits, but this technique is not compatible with qubits being controlled. By decoupling a spin qubit from low-frequency noise using feedback, we engineer the qubit’s coherence and investigate its effects on control fidelity. We find that for feedback-protected single-electron spin, the control fidelity is eventually limited by high-frequency charge noise.

By employing quantum dots fabricated in a gallium arsenide semiconductor, we use the qubit itself to estimate the instantaneous noise in real time. We actively compensate for the noise by adjusting microwave control pulses. The compensation suppresses low-frequency noise (below 1 kHz) and boosts the qubit coherence time to 766 ns (from 28 ns without compensation) and the control fidelity to 99%.

While the coherence time is limited by remaining low-frequency noise, the decay of the microwave-driven qubit is due to noise in the megahertz range, which we identify as noise from charges that tunnel into or out from impurities in the material.

Our real-time coherence-protection methods promise scalable solutions for quantum devices. Our findings on the noise effect will accelerate further improvements of device structures and materials toward spin-based quantum computers.

Figure 1

Example of noise power spectra for spin qubits with and without feedback. A typical noise spectrum composed of $1/f^2$ and $1/f$ noise is shown in a log-log plot (black line). The feedback control acts like an active filter suppressing the low-frequency noise (red line). Shown on the bottom are relevant frequencies with $\mathrm{\Delta }t$ the feedback latency, $t$ the qubit evolution time at which the coherence is evaluated, and $f_{\text{rabi}}$ the Rabi frequency.

Figure 2

Ramsey measurement and feedback-control scheme of an electron spin qubit. (a) False-colored scanning electron micrograph image of the TQD device. An electron spin qubit in the middle QD (red arrow with a circle) is controlled by the EDSR where the spin is coupled to a microwave (MW) electric field via a stray magnetic field of the micromagnet deposited on the wafer surface [19]. The right QD hosts an electron spin (blue arrow with a circle) used as a readout ancilla, while the left QD hosts another electron which is unused and decoupled from the two spins. The energy detuning between the middle and the right QDs ($ϵ$) is gate tunable, and the QD electron occupancies are probed by a proximal single-electron transistor [23]. (b) Schematic of the Ramsey measurement. Two electrons (qubit and ancilla) are initialized to a doubly occupied singlet state in the right QD, and an up-spin qubit is prepared by adiabatically loading one of the electrons to the middle QD [22]. Two $\pi /2$ microwave bursts, separated by time $t_R$, are applied (before and between these bursts, off-resonant microwave bursts are optionally applied). The ancilla-spin state is not affected by the microwave bursts. The final state is read out by unloading an up-spin (antiparallel to the ancilla) state from the middle QD, while a down-spin (parallel to the ancilla) state remains blocked. (c) Up-spin probability $P_{\uparrow }$ as a function of $t_R$. The lower panel shows the Ramsey oscillations whose frequency varies with the laboratory time due to Overhauser field fluctuations. Each data point of $P_{\uparrow }$ is calculated from 100 single-shot readout outcomes. The upper panel shows the trace obtained by averaging all the oscillations in the lower panel. The decay envelope gives the dephasing time of $T_2^{*}=28.4\mathrm{ns}$, a value typical for electron spins in GaAs heterostructures. (d) Schematic of the feedback-control loop for a spin qubit. Data of a Ramsey oscillation as shown in (c) are processed in a digital signal processing (DSP) hardware with programmable logic (FPGA) to estimate the frequency detuning $\delta f=f_{\text{qubit}}-f_{\text{qubit}}^{\text{est}}$ between the current qubit frequency $f_{\text{qubit}}$ and its previous estimate $f_{\text{qubit}}^{\text{est}}$ (“probe” step). The value of $f_{\text{qubit}}^{\text{est}}$ is updated to $f_{\text{qubit}}^{\text{est}}\mapsto f_{\text{qubit}}^{\text{est}}+\delta f$ (“update” step), after which the target experiment follows (“target” step). In the ideal case, the subsequent qubit algorithms can be executed with a microwave frequency $f_{\mathrm{MW}}$ matching $f_{\text{qubit}}$ exactly (by choosing $\mathrm{\Delta }=0$).

Figure 3

Suppressed dephasing of an electron spin qubit in a feedback-controlled rotating frame. (a) Time dependence of the frequency detuning $\delta f=f_{\text{qubit}}-f_{\text{qubit}}^{\text{est}}$ extracted from Ramsey measurements. The blue trace is taken with a fixed $f_{\mathrm{MW}}$, and the red trace is taken with a feedback-controlled $f_{\mathrm{MW}}$. The inset shows the histograms of $\delta f$ in the two cases. The histogram with the feedback control exhibits a normal distribution with a variance ${\sigma }^2=⟨(\delta f{)}^2⟩=(0.294\mathrm{MHz}{)}^2$ (black dashed curve). (b) Ramsey oscillations as in Fig. 2 but obtained with the feedback control ($\mathrm{\Delta }=50\mathrm{MHz}$). The envelope of the oscillation in the upper panel is a Gaussian decay function drawn using dephasing time $T_2^{*}=1/(\pi \sqrt{2}\sigma )=766.7\mathrm{ns}$. (c) Variance of $\delta f$, ${\sigma }^2=⟨(\delta f{)}^2⟩$, as a function of the latency $\mathrm{\Delta }t$ between the probe and target steps (orange circles) and that of the frequency correlator ${\sigma }_B^2$ in the laboratory frame (blue circles) as a function of the time difference $\mathrm{\Delta }t$. A blue line is a fit to ${\sigma }_B^2=D\mathrm{\Delta }{t}^{\alpha }$ and shows subdiffusive behavior with the exponent $\alpha =0.84$ similar to a value found for singlet-triplet oscillations [27]. The orange curve is a fit to ${\sigma }^2=D\mathrm{\Delta }{t}^{\alpha }+(0.288\mathrm{MHz}{)}^2$.

Figure 4

Coherent control and benchmarking of a single-electron spin with the feedback. (a) Typical Rabi oscillations versus the offset $\mathrm{\Delta }$ obtained in the feedback-controlled rotating frame. A horizontal white dashed line indicates the microwave-induced shift of the qubit frequency, $\mathrm{\Delta }{f}_{\text{qubit}}$. (b) Microwave-amplitude dependence of the qubit frequency shift. A blue line is a fit to the data showing that the amplitude dependence is quadratic ($\propto E^{1.95\pm 0.18}$). (c) Rabi oscillations obtained at zero detuning upon compensating for the induced shift $\mathrm{\Delta }{f}_{\text{qubit}}$. Namely, an off-resonant microwave burst with the same amplitude as the one used for the Rabi measurement is applied before and during the Ramsey measurement [see Fig. 2]. A similar off-resonant burst is also applied for 200 ns before the Rabi drive of duration $t_b$ to stabilize the value of the frequency shift. (d) Microwave-amplitude dependence of the Rabi frequency (blue circles) and the Rabi decay time (red squares with lines). A blue line is a linear fit to the low-amplitude data of the Rabi frequency. (e) Normalized sequence fidelities for standard (top) and interleaved (others) randomized benchmarking. Traces are offset by 0.4 for clarity. The standard sequence shows an exponential decay with an average single-gate fidelity of $97.50\pm 0.05\%$. Interleaved sequences are annotated with corresponding single-qubit gates and extracted fidelities.

Figure 5

Rabi noise spectroscopy. (a) Power spectral density $S(f)$ of the longitudinal noise in $f_{\text{qubit}}$ (red circles) extracted from the data in Fig. 4. Vertical gray lines show Larmor precession frequencies for three nuclear species (${}^{75}\mathrm{As}$, ${}^{69}\mathrm{Ga}$, and ${}^{71}\mathrm{Ga}$) calculated with the micromagnet-induced field component of $B_z^{\mathrm{MM}}=70\mathrm{mT}$. Black and green lines are guides to the eye for $f^{-1}$ and $f^2$ dependence, respectively. The inset illustrates electron-nuclear spin coupling in an inhomogeneous magnetic field. Each nuclear spin is randomly oriented and precesses around a local magnetic field vector $\mathit{B}({\mathit{r}}_k)$, leading to an oscillatory Overhauser field in the $z$ direction. (b) Comparison of $S(f)$ in (a) and those extracted from Ramsey measurements. The power spectral density of $\delta f$ in Ramsey measurements is calculated by the fast Fourier transform of $\delta f(t)$. The spectral density taken with small SEC and feedback off (blue) shows the $f^{-1.7}$ dependence (black dashed line) similar to those observed for nuclear spin diffusion noise [15, 16]. The spectral density is significantly suppressed with feedback on (orange) down to a level determined by the precision of the feedback control. The flat noise spectrum suggests that the residual low-frequency noise is uncorrelated. A peak at 2 Hz is due to the vibration of the dilution refrigerator. The power spectral density increases as the SEC is increased (red curve in the top left), and it follows the $f^{-1}$ dependence (black solid line) in line with $S(f)$ extracted from the Rabi spectroscopy in (a) (red curve in the bottom right).

Figure 6

Coherent oscillations with a large SEC. These data are taken in a gate-bias condition similar to those of Figs. 4 and 5, where the SEC is increased. (a) Ramsey oscillation. Spin-up probability $P_{\uparrow }$ is measured as a function of $t_R$ with the feedback on ($\mathrm{\Delta }=30\mathrm{MHz}$). The solid curve is a fit to the Gaussian-decaying oscillation with $T_2^{*}=103.7\mathrm{ns}$. (b) Rabi oscillation for $Q=85$. The solid curve is a fit to the exponentially decaying oscillation with $f_{\text{rabi}}=33.64\pm 0.01\mathrm{MHz}$ and $T_2^{\text{rabi}}=1.26\pm 0.12\mu \mathrm{s}$.

Figure 7

Microwave amplitude dependence of the Rabi frequency (a) and Rabi noise spectroscopy (b) with different SECs. Red circles are taken in the “large SEC” condition with $A=0.6\mathrm{MHz}$ as shown in Fig. 5. Blue squares are taken in a condition where the SEC is increased to $A=1.0\mathrm{MHz}$ and the $Q$ factor is slightly lower.